Summary: Journd of Pure and Applied Algebra 40 (1986) 103-113
North-Holland
103
GLOBAL OF CROSSED
Eli ALJADEFF and Shmuel ROSSET
Tel Aviv University, Ramat Aviv, 69978 Israel
Communicated by H. Bass
Received 18 December 1984
Revised 22 April 1985
Suppose r is a group and a homomorphism t : I'-+ Aut(K) is given. Here K is a
field and Aut(K) is the group of field automorphisms of K. Then we say that r acts
on K. In such circumstances the multiplicative group K* is a r-module and it is well
known that elements of H2(r, K*) give rise to `crossed product' algebras. To recall
this let aeN2(r, K*) and let f: TxT-+K* be a 2-cocycle representing a. One
defines the crossed product, which we denote by
as follows. As left K vector space it is a direct sum UaErKuo. Multiplication is
defined so as to satisfy the rule
Here a(~) is the action of t(a) on y. It is easy to see that this multiplication is
associative (this follows from the cocycle condition) and that, up to isomorphism
of rings, KfT only depends on (Y, not on the choice of f. It is thus assumed